Best Known (84, 84+19, s)-Nets in Base 4
(84, 84+19, 1822)-Net over F4 — Constructive and digital
Digital (84, 103, 1822)-net over F4, using
- 43 times duplication [i] based on digital (81, 100, 1822)-net over F4, using
- net defined by OOA [i] based on linear OOA(4100, 1822, F4, 19, 19) (dual of [(1822, 19), 34518, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4100, 16399, F4, 19) (dual of [16399, 16299, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4100, 16400, F4, 19) (dual of [16400, 16300, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(499, 16385, F4, 19) (dual of [16385, 16286, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(485, 16385, F4, 17) (dual of [16385, 16300, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(41, 15, F4, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4100, 16400, F4, 19) (dual of [16400, 16300, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4100, 16399, F4, 19) (dual of [16399, 16299, 20]-code), using
- net defined by OOA [i] based on linear OOA(4100, 1822, F4, 19, 19) (dual of [(1822, 19), 34518, 20]-NRT-code), using
(84, 84+19, 9786)-Net over F4 — Digital
Digital (84, 103, 9786)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4103, 9786, F4, 19) (dual of [9786, 9683, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4103, 16404, F4, 19) (dual of [16404, 16301, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- linear OA(499, 16384, F4, 19) (dual of [16384, 16285, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(478, 16384, F4, 15) (dual of [16384, 16306, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(41, 17, F4, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4103, 16404, F4, 19) (dual of [16404, 16301, 20]-code), using
(84, 84+19, large)-Net in Base 4 — Upper bound on s
There is no (84, 103, large)-net in base 4, because
- 17 times m-reduction [i] would yield (84, 86, large)-net in base 4, but